Topological Hochschild homology of l and ko

We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum.

💡 Research Summary

The paper undertakes a complete calculation of the integral homotopy groups of the topological Hochschild homology (THH) of two fundamental connective K‑theory spectra: the Adams summand ℓ (the p‑local connective complex K‑theory summand) for an arbitrary prime p, and the connective real K‑theory spectrum ko at the prime p = 2. THH, as a spectrum‑level analogue of Hochschild homology, plays a central role in modern algebraic K‑theory, especially through its relationship with topological cyclic homology (TC) and higher trace methods. However, explicit computations of THH are notoriously difficult because they involve intricate spectral sequences, torsion phenomena, and higher operations.

The authors begin by revisiting Bökstedt’s original computation of THH(ℤ) and the Bökstedt‑Hsiang‑Madsen analysis of THH(ℤ/p). They adapt these techniques to the Adams summand ℓ, exploiting the fact that ℓ is a p‑complete ℤₚ‑module equipped with a well‑behaved action of the Adams operations. By constructing a Bökstedt‑Madsen type spectral sequence whose E₂‑page reflects the ℓ‑module structure, they are able to identify a clear filtration by degree. The resulting homotopy groups decompose as a direct sum of free ℤₚ‑modules in even degrees and ℤ/p‑torsion in odd degrees: specifically, πₙ THH(ℓ) ≅ Σ^{2k}ℤₚ ⊕ Σ^{2k+1}ℤ/p for non‑negative integers k. This pattern mirrors the periodic Bott element in ℓ and demonstrates that the p‑adic periodicity of ℓ is faithfully reflected in its THH.

The second major part of the work focuses on ko at the prime 2. The connective real K‑theory spectrum carries a richer 2‑adic structure and a more complicated module action, making the THH calculation substantially more involved. The authors first analyze the module structure of ko over the sphere spectrum and then set up a Bökstedt‑Madsen spectral sequence adapted to the 2‑adic context. By carefully tracking the action of the 2‑primary Adams operations and the resulting differentials, they determine that the homotopy groups of THH(ko) exhibit a 4‑periodic pattern: πₙ THH(ko) ≅ Σ^{4k}ℤ₂ in degrees n = 4k and πₙ THH(ko) ≅ Σ^{4k+2}ℤ/2 in degrees n = 4k + 2. Thus, even degrees carry 2‑complete free modules, while the intermediate degrees carry pure 2‑torsion. This alternating structure aligns with the 8‑fold Bott periodicity of real K‑theory and provides a concrete description of the 2‑adic torsion that appears in higher algebraic K‑theory via trace methods.

Beyond the raw calculations, the paper discusses the implications for related invariants. The THH of ℓ and ko feed directly into the computation of TC(ℓ) and TC(ko), respectively, because the cyclotomic structure on THH is the input for the TC construction. The authors show that the periodicities observed in THH match the expected periodicities in TC after taking homotopy fixed points and Tate constructions. Moreover, the explicit description of torsion components clarifies how the 2‑adic and p‑adic information in algebraic K‑theory is organized, offering a bridge between the homotopy‑theoretic calculations and more algebraic approaches such as the Dundas–Goodwillie–McCarthy theorem.

In conclusion, the paper delivers a thorough, integral-level computation of THH for two cornerstone spectra in stable homotopy theory. The results not only provide concrete algebraic models for these THH groups but also lay the groundwork for future work on higher algebraic K‑theory, trace methods, and the interplay between real and complex K‑theory at the level of topological Hochschild and cyclic homology.